Subject: Chemistry — Solid State
Question: For the give close packed structure of a salt made of
cation X and anion Y shown below (ions of only one face are shown for
clarity), the packing fraction is
approximately packing fraction $= \frac{packing\ efficiency}{100}$
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Answer: 0.63
Solution: $2\left\lbrack r_{x} + r_{y} \right\rbrack = \sqrt{2}a$
$\left\{ a = 2r_{y};r_{y} = 0.59 \right\}$
$$2\left\lbrack r_{x} + r_{y} \right\rbrack = \sqrt{2}\left( 2r_{y} \right)$$
$$2r_{x} = 2\sqrt{2}r_{y} - 2r_{y}$$
$$r_{x} = r_{y}\left\lbrack \sqrt{2} - 1 \right\rbrack$$
$$r_{x} = 0.414,r_{y} = 0.207\ a$$
Packing fraction
$= \frac{3 \times \frac{4}{3}\pi r_{x}^{3} + \frac{4}{3}\pi r_{y}^{3}}{a^{3}} = 0.63$